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Spinon Fractionalization from Dynamic Structure Factor of Spin-1 Spinon Fractionalization from Dynamic Structure Factor of Spin-1/2 Heisenberg Antiferromagnet on the Kagome Lattice W. Zhu1, Shou-Shu Gong2,3, D. N. Sheng3 1Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Department of Physics, Beihang University, Beijing, 100191, China 3Department of Physics and Astronomy, California State University, Northridge, California 91330, USA We study dynamical spin structure factor (DSSF) of S = 1/2 Heisenberg model on the kagome lattice (KAFM) by means of density-matrix renormalization group. By comparison with the well-defined magnetic ordered state and chiral spin liquid sitting nearby in the phase diagram, the KAFM with the nearest-neighbor interaction shows distinct dynamical response behaviors. First of all, the DSSF displays important spectral intensity predominantly at low frequency region around Q = M point in momentum space, and shows a broad spectral distribution at high frequency region for momenta along the boundary of the extended Brillouin zone. Secondly, spinon excitation spectrum is identified from momentum and energy resolved DSSF, which shows critical behavior with much reduced spectrum intensity comparing to the neighboring chiral spin liquid. By adding a weak Dzyaloshinkii-Moriya interaction, the DSSF demonstrates a strong sensitivity to the boundary conditions more consistent with a gapless spin liquid. These results capture the main observations in the inelastic neutron scattering measurements of herbertsmithite, and indicate the spin liquid nature of the ground state with fractionalized spinon excitations. By following the DSSF crossing the quantum phase transition between the CSL and the magnetical ordered phase, we identify the spinon condensation driving the quantum phase transition. PACS numbers: 75.10.Jm, 75.40.Mg, 75.40.Gb Introduction.— Quantum spin liquid (QSL) is a novel quan- 31], which captures the dominant interaction for the herbert- tum phase which behaves differently from conventional mag- smithite. Although a QSL ground state has been established netic states [1–3]. It does not show any magnetic order or in the KAFM, the full nature of the QSL including the na- lattice symmetry breaking even approaching zero temperature ture of the fractionalized quasi-particles, and the existence of limit. Theoretical studies have shown the intrinsic nature of a spin gap is still under debate. While earlier density ma- QSL including massive entanglement and fractionalized exci- trix renormalization group (DMRG) simulation suggested a tations [4–7], which are challenging to be measured directly gapped spin liquid [24–27], parton construction and varia- in experiments. Experimentally, QSL candidates have been tional Monte Carlo study found the optimized ground state identified in frustrated magnetic materials such as kagome- as the gapless U(1) Dirac spin liquid [19–22]. Such a gap- and triangular-lattice compounds [8–15]. The kagome anti- less spin liquid scenario is also indirectly supported by recent ferromagnet herbertsmithite [8–11] is one of the most promis- DMRG targeting the system response to the inserted flux and ing spin-liquid candidates. The magnetic order of the material tensor network results [32, 33]. However, more direct evi- has been excluded down to temperatures a few orders [9, 10] dence from low energy excitations are still absent. The open of magnitude below coupling energy scale. Furthermore, the question regarding the nature of the QSL phase demands new inelastic neutron scattering (INS) measurement characterizes theoretical approaches beyond the ground state study such the dynamic spin structure factor (DSSF) S(Q,ω) as a broad as the DSSF that reveals excitation properties. So far most continuumspectrum in higher frequency regime [10], which is of the studies on the DSSF of the KAFM are based on the distinctly different from the spectrum of conventional magnon mean-field analysis or approximate methods [34–38], the un- excitations. It remains an open issue what information regard- biased numerical model calculation is rare limited to small ing topological order of the state, and the fractionalization of systems [39]. spin excitations can be extracted from such measurements. In In this paper, we aim to understand the DSSF based arXiv:1804.03333v1 [cond-mat.str-el] 10 Apr 2018 particular, some other factors in material such as disorder may on large-scale DMRG for the KAFM and extended models also lead to a similar continuum of S(Q,ω) [16], making the- with either small further-neighbor Heisenberg interactions or oretical understanding of the INS essential for distinguishing Dzyaloshinskii-Moriya (DM) interaction, which are both rel- different physics. In addition, whether such a spin-liquid can- evant to experimental material. With these pertubative cou- didate has a finite spin gap remains unresolved in experimen- plings, we identify characteristic features of the DSSF for dif- tal probes including the INS [10] and the nuclear magnetic ferent quantum phases, including a q = (0, 0) magnetic order resonance (NMR) [11]. To clarify these questions, theoreti- phase, a gapped chiral spin liquid (CSL), and a QSL connect- cal studies on dynamic measurements related to experimental ing to the phase of the pure KAFM (we denote it as KSL). In probes are highly desired. the q = (0, 0) phase, the key signature of long-rangemagnetic In theoretical study, the ground state of the spin-1/2 order is the appearanceof sharp gapless dispersive modes with kagome antiferromagnet (KAFM) with nearest-neighbor the largest intensity at the corresponding magnetic wave vec- Heisenberg interaction has been investigated intensively [17– tor. In the CSL phase, the energy scans of the DSSF show 2 (a) (d) (b) (c) S(q ,q ) CSL x y 2.400 0.2 2.300 VBC 2.200 2.100 2.000 1.900 1.800 3 1.700 J 1.600 1.500 1.400 1.300 1.200 1.100 0.1 1.000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 KSL q = (0,0) 0.000 0 0 0.05 0.1 0.15 0.2 0.25 J2 FIG. 1. Static spin structure factor of the kagome model in different quantum phases. (a) Quantum phase diagram of the kagome model in the J2 − J3 plane obtained in Ref. [40]. (b-d) are static spin structure factor in momentum space for (b) the q = (0, 0) phase at J2 = 0.25, J3 = 0.0, (c) the CSL at J2 = 0.25, J3 = 0.25, and (d) the KSL at J2 = J3 = 0. The extended Brillouin zone is marked by the white dashed line. intensity peak at finite frequency, which illustrates the emer- as gent gapped spinon pair excitations. In the KSL, the momen- z z 1 iQ·(ri−rj ) z z tum resolved DSSF concentrates along the boundary of the S(Q)= hS (−Q)S (Q)i = e hSi Sj i, N X extended Brillouin zone (BZ) and shows a broad maximum at i,j the M point, which are consistent with the INS results of the where the wave vector Q ~ ~ in the herbertsmithite. In the energy scans for the KSL, the intensity = (q1, q2) = q1b1 + q2b2 ~ of the DSSF forms a continuum, which extends over a wide BZ is defined by reciprocal vectors b1,2 (see Fig. 2(d)). In frequency range, concomitant with a pronounced intensity at Fig. 1(b), S(Q) shows sharp peaks at the M points, showing low energy region. The evidences from DSSF, including the a q = (0, 0) magnetic order [45]. In the nonmagnetic phases fractionalized spinon continuum in energy scans, the sensitiv- S(Q) is featureless as shown in Fig. 1(c-d). In the KSL phase, ity of excitation gap by imposing different boundary condi- S(Q) concentrates along the boundary of the extended BZ tions (BCs), and by tuning DM perturbation, are in support of and shows broad maximum near the M point, which agree a QSL with gapless fractionalized spin excitations. with the features of the INS data of herbertsmithite [10]. Dynamic spin structure factor.— The DSSF is defined as Model and Method.— We study the spin-1/2 KAFM with further-neighbor antiferromagnetic interactions 1 1 Sαβ(Q,ω)= − ImhSα(−Q) Sβ(Q)i, π ω − (H − E0)+ iη H = J1 X Si ·Sj +J2 X Si ·Sj +J3 X Si ·Sj , (1) where E0 is the ground-state energy, η → 0 is a small smear- hi,ji hhi,jii hhhi,jiii ing energy [44], and α, β denote spin components. First of all, we discuss the salient features of the DSSF in different phases as shown in Fig. 2. For the q phase (Fig. 2(a,d)), where J ,J ,J are the first-, second-, and third-neighbor = (0, 0) 1 2 3 we observe a sharp peak at the point with , serving couplings (J is the coupling inside the hexagon and we take M ω = 0 3 as the key signature of the long-range magnetic order with J1 =1 as the energy scale). The previously obtained DMRG phase diagram is shown in Fig. 1(a) [40]. Different neighbor the largest intensity at ordering wave vector. Interestingly, we also observe a small peak with a broader and reduced phases surround the KSL sitting near the J point, including a 1 weight in higher energy region, which we speculate related to q = (0, 0) magnetic order phase, a gapped CSL phase, and a valence-bond solid phase. two-magnon excitations. For all other momenta, the intensity shows broad distribution in the energy scans. In this study, we develop a DMRG program to calculate In the CSL, the DSSF along the high-symmetricline is pre- dynamic structure factor [41–43], which can apply to general sented in Fig. 2(b,e), showing a fully gapped excitation branch strongly correlated systems.
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